Equivalent conditions and applications of a class of Hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernels

• Main Authors: Junfei Cao, Bing He, Yong Hong, Bicheng Yang
• Format: Article
• Language: English
• Published: SpringerOpen 2018-08-01
• Series: Journal of Inequalities and Applications
• Subjects: Hilbert-type integral inequality; Quasi-homogeneous kernel; Equivalent conditions; Best constant factor
• Online Access: http://link.springer.com/article/10.1186/s13660-018-1797-5

Abstract Let K(x1,…,xn) $K(x_{1},\ldots,x_{n})$ satisfy K(x1,…,txi,…,xn)=tλλiK(t−λiλ1x1,…,xi,…,t−λiλnxn) $$K(x_{1},\ldots,tx_{i},\ldots,x_{n})=t^{\lambda\lambda _{i}}K \bigl(t^{-\frac{\lambda_{i}}{\lambda_{1}}}x_{1},\ldots,x_{i}, \ldots,t^{-\frac {\lambda _{i}}{\lambda_{n}}}x_{n} \bigr) $$ for t>0 $t>0$. With this integral kernel, by using the method and technique of weight coefficients, the equivalent conditions and the best constant factors for the validity of Hilbert-type integral inequalities involving multiple functions are discussed. Finally, the applications of the integral inequalities are considered.